3.77 \(\int \frac{x^2 \log (e (\frac{a+b x}{c+d x})^n)}{f-g x^2} \, dx\)

Optimal. Leaf size=550 \[ \frac{\sqrt{f} n \text{PolyLog}\left (2,-\frac{\sqrt{g} (a+b x)}{b \sqrt{f}-a \sqrt{g}}\right )}{2 g^{3/2}}-\frac{\sqrt{f} n \text{PolyLog}\left (2,\frac{\sqrt{g} (a+b x)}{a \sqrt{g}+b \sqrt{f}}\right )}{2 g^{3/2}}-\frac{\sqrt{f} n \text{PolyLog}\left (2,-\frac{\sqrt{g} (c+d x)}{d \sqrt{f}-c \sqrt{g}}\right )}{2 g^{3/2}}+\frac{\sqrt{f} n \text{PolyLog}\left (2,\frac{\sqrt{g} (c+d x)}{c \sqrt{g}+d \sqrt{f}}\right )}{2 g^{3/2}}-\frac{\sqrt{f} \tanh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \left (-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )}{g^{3/2}}+\frac{x \left (-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )}{g}-\frac{\sqrt{f} n \log (a+b x) \log \left (\frac{b \left (\sqrt{f}-\sqrt{g} x\right )}{a \sqrt{g}+b \sqrt{f}}\right )}{2 g^{3/2}}+\frac{\sqrt{f} n \log (a+b x) \log \left (\frac{b \left (\sqrt{f}+\sqrt{g} x\right )}{b \sqrt{f}-a \sqrt{g}}\right )}{2 g^{3/2}}-\frac{n (a+b x) \log (a+b x)}{b g}+\frac{\sqrt{f} n \log (c+d x) \log \left (\frac{d \left (\sqrt{f}-\sqrt{g} x\right )}{c \sqrt{g}+d \sqrt{f}}\right )}{2 g^{3/2}}-\frac{\sqrt{f} n \log (c+d x) \log \left (\frac{d \left (\sqrt{f}+\sqrt{g} x\right )}{d \sqrt{f}-c \sqrt{g}}\right )}{2 g^{3/2}}+\frac{n (c+d x) \log (c+d x)}{d g} \]

[Out]

-((n*(a + b*x)*Log[a + b*x])/(b*g)) + (n*(c + d*x)*Log[c + d*x])/(d*g) + (x*(n*Log[a + b*x] - Log[e*((a + b*x)
/(c + d*x))^n] - n*Log[c + d*x]))/g - (Sqrt[f]*ArcTanh[(Sqrt[g]*x)/Sqrt[f]]*(n*Log[a + b*x] - Log[e*((a + b*x)
/(c + d*x))^n] - n*Log[c + d*x]))/g^(3/2) - (Sqrt[f]*n*Log[a + b*x]*Log[(b*(Sqrt[f] - Sqrt[g]*x))/(b*Sqrt[f] +
a*Sqrt[g])])/(2*g^(3/2)) + (Sqrt[f]*n*Log[c + d*x]*Log[(d*(Sqrt[f] - Sqrt[g]*x))/(d*Sqrt[f] + c*Sqrt[g])])/(2
*g^(3/2)) + (Sqrt[f]*n*Log[a + b*x]*Log[(b*(Sqrt[f] + Sqrt[g]*x))/(b*Sqrt[f] - a*Sqrt[g])])/(2*g^(3/2)) - (Sqr
t[f]*n*Log[c + d*x]*Log[(d*(Sqrt[f] + Sqrt[g]*x))/(d*Sqrt[f] - c*Sqrt[g])])/(2*g^(3/2)) + (Sqrt[f]*n*PolyLog[2
, -((Sqrt[g]*(a + b*x))/(b*Sqrt[f] - a*Sqrt[g]))])/(2*g^(3/2)) - (Sqrt[f]*n*PolyLog[2, (Sqrt[g]*(a + b*x))/(b*
Sqrt[f] + a*Sqrt[g])])/(2*g^(3/2)) - (Sqrt[f]*n*PolyLog[2, -((Sqrt[g]*(c + d*x))/(d*Sqrt[f] - c*Sqrt[g]))])/(2
*g^(3/2)) + (Sqrt[f]*n*PolyLog[2, (Sqrt[g]*(c + d*x))/(d*Sqrt[f] + c*Sqrt[g])])/(2*g^(3/2))

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Rubi [A]  time = 0.565428, antiderivative size = 550, normalized size of antiderivative = 1., number of steps used = 27, number of rules used = 10, integrand size = 32, \(\frac{\text{number of rules}}{\text{integrand size}}\) = 0.312, Rules used = {2513, 321, 208, 2416, 2389, 2295, 2409, 2394, 2393, 2391} \[ \frac{\sqrt{f} n \text{PolyLog}\left (2,-\frac{\sqrt{g} (a+b x)}{b \sqrt{f}-a \sqrt{g}}\right )}{2 g^{3/2}}-\frac{\sqrt{f} n \text{PolyLog}\left (2,\frac{\sqrt{g} (a+b x)}{a \sqrt{g}+b \sqrt{f}}\right )}{2 g^{3/2}}-\frac{\sqrt{f} n \text{PolyLog}\left (2,-\frac{\sqrt{g} (c+d x)}{d \sqrt{f}-c \sqrt{g}}\right )}{2 g^{3/2}}+\frac{\sqrt{f} n \text{PolyLog}\left (2,\frac{\sqrt{g} (c+d x)}{c \sqrt{g}+d \sqrt{f}}\right )}{2 g^{3/2}}-\frac{\sqrt{f} \tanh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \left (-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )}{g^{3/2}}+\frac{x \left (-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+n \log (a+b x)-n \log (c+d x)\right )}{g}-\frac{\sqrt{f} n \log (a+b x) \log \left (\frac{b \left (\sqrt{f}-\sqrt{g} x\right )}{a \sqrt{g}+b \sqrt{f}}\right )}{2 g^{3/2}}+\frac{\sqrt{f} n \log (a+b x) \log \left (\frac{b \left (\sqrt{f}+\sqrt{g} x\right )}{b \sqrt{f}-a \sqrt{g}}\right )}{2 g^{3/2}}-\frac{n (a+b x) \log (a+b x)}{b g}+\frac{\sqrt{f} n \log (c+d x) \log \left (\frac{d \left (\sqrt{f}-\sqrt{g} x\right )}{c \sqrt{g}+d \sqrt{f}}\right )}{2 g^{3/2}}-\frac{\sqrt{f} n \log (c+d x) \log \left (\frac{d \left (\sqrt{f}+\sqrt{g} x\right )}{d \sqrt{f}-c \sqrt{g}}\right )}{2 g^{3/2}}+\frac{n (c+d x) \log (c+d x)}{d g} \]

Antiderivative was successfully verified.

[In]

Int[(x^2*Log[e*((a + b*x)/(c + d*x))^n])/(f - g*x^2),x]

[Out]

-((n*(a + b*x)*Log[a + b*x])/(b*g)) + (n*(c + d*x)*Log[c + d*x])/(d*g) + (x*(n*Log[a + b*x] - Log[e*((a + b*x)
/(c + d*x))^n] - n*Log[c + d*x]))/g - (Sqrt[f]*ArcTanh[(Sqrt[g]*x)/Sqrt[f]]*(n*Log[a + b*x] - Log[e*((a + b*x)
/(c + d*x))^n] - n*Log[c + d*x]))/g^(3/2) - (Sqrt[f]*n*Log[a + b*x]*Log[(b*(Sqrt[f] - Sqrt[g]*x))/(b*Sqrt[f] +
a*Sqrt[g])])/(2*g^(3/2)) + (Sqrt[f]*n*Log[c + d*x]*Log[(d*(Sqrt[f] - Sqrt[g]*x))/(d*Sqrt[f] + c*Sqrt[g])])/(2
*g^(3/2)) + (Sqrt[f]*n*Log[a + b*x]*Log[(b*(Sqrt[f] + Sqrt[g]*x))/(b*Sqrt[f] - a*Sqrt[g])])/(2*g^(3/2)) - (Sqr
t[f]*n*Log[c + d*x]*Log[(d*(Sqrt[f] + Sqrt[g]*x))/(d*Sqrt[f] - c*Sqrt[g])])/(2*g^(3/2)) + (Sqrt[f]*n*PolyLog[2
, -((Sqrt[g]*(a + b*x))/(b*Sqrt[f] - a*Sqrt[g]))])/(2*g^(3/2)) - (Sqrt[f]*n*PolyLog[2, (Sqrt[g]*(a + b*x))/(b*
Sqrt[f] + a*Sqrt[g])])/(2*g^(3/2)) - (Sqrt[f]*n*PolyLog[2, -((Sqrt[g]*(c + d*x))/(d*Sqrt[f] - c*Sqrt[g]))])/(2
*g^(3/2)) + (Sqrt[f]*n*PolyLog[2, (Sqrt[g]*(c + d*x))/(d*Sqrt[f] + c*Sqrt[g])])/(2*g^(3/2))

Rule 2513

Int[Log[(e_.)*((f_.)*((a_.) + (b_.)*(x_))^(p_.)*((c_.) + (d_.)*(x_))^(q_.))^(r_.)]*(RFx_.), x_Symbol] :> Dist[
p*r, Int[RFx*Log[a + b*x], x], x] + (Dist[q*r, Int[RFx*Log[c + d*x], x], x] - Dist[p*r*Log[a + b*x] + q*r*Log[
c + d*x] - Log[e*(f*(a + b*x)^p*(c + d*x)^q)^r], Int[RFx, x], x]) /; FreeQ[{a, b, c, d, e, f, p, q, r}, x] &&
RationalFunctionQ[RFx, x] && NeQ[b*c - a*d, 0] &&  !MatchQ[RFx, (u_.)*(a + b*x)^(m_.)*(c + d*x)^(n_.) /; Integ
ersQ[m, n]]

Rule 321

Int[((c_.)*(x_))^(m_)*((a_) + (b_.)*(x_)^(n_))^(p_), x_Symbol] :> Simp[(c^(n - 1)*(c*x)^(m - n + 1)*(a + b*x^n
)^(p + 1))/(b*(m + n*p + 1)), x] - Dist[(a*c^n*(m - n + 1))/(b*(m + n*p + 1)), Int[(c*x)^(m - n)*(a + b*x^n)^p
, x], x] /; FreeQ[{a, b, c, p}, x] && IGtQ[n, 0] && GtQ[m, n - 1] && NeQ[m + n*p + 1, 0] && IntBinomialQ[a, b,
c, n, m, p, x]

Rule 208

Int[((a_) + (b_.)*(x_)^2)^(-1), x_Symbol] :> Simp[(Rt[-(a/b), 2]*ArcTanh[x/Rt[-(a/b), 2]])/a, x] /; FreeQ[{a,
b}, x] && NegQ[a/b]

Rule 2416

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((h_.)*(x_))^(m_.)*((f_) + (g_.)*(x_)^(r_.))^(q
_.), x_Symbol] :> Int[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (h*x)^m*(f + g*x^r)^q, x], x] /; FreeQ[{a,
b, c, d, e, f, g, h, m, n, p, q, r}, x] && IntegerQ[m] && IntegerQ[q]

Rule 2389

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.), x_Symbol] :> Dist[1/e, Subst[Int[(a + b*Log[c*
x^n])^p, x], x, d + e*x], x] /; FreeQ[{a, b, c, d, e, n, p}, x]

Rule 2295

Int[Log[(c_.)*(x_)^(n_.)], x_Symbol] :> Simp[x*Log[c*x^n], x] - Simp[n*x, x] /; FreeQ[{c, n}, x]

Rule 2409

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))^(p_.)*((f_) + (g_.)*(x_)^(r_))^(q_.), x_Symbol] :> In
t[ExpandIntegrand[(a + b*Log[c*(d + e*x)^n])^p, (f + g*x^r)^q, x], x] /; FreeQ[{a, b, c, d, e, f, g, n, r}, x]
&& IGtQ[p, 0] && IntegerQ[q] && (GtQ[q, 0] || (IntegerQ[r] && NeQ[r, 1]))

Rule 2394

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))^(n_.)]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Simp[(Log[(e*(f +
g*x))/(e*f - d*g)]*(a + b*Log[c*(d + e*x)^n]))/g, x] - Dist[(b*e*n)/g, Int[Log[(e*(f + g*x))/(e*f - d*g)]/(d +
e*x), x], x] /; FreeQ[{a, b, c, d, e, f, g, n}, x] && NeQ[e*f - d*g, 0]

Rule 2393

Int[((a_.) + Log[(c_.)*((d_) + (e_.)*(x_))]*(b_.))/((f_.) + (g_.)*(x_)), x_Symbol] :> Dist[1/g, Subst[Int[(a +
b*Log[1 + (c*e*x)/g])/x, x], x, f + g*x], x] /; FreeQ[{a, b, c, d, e, f, g}, x] && NeQ[e*f - d*g, 0] && EqQ[g
+ c*(e*f - d*g), 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rubi steps

\begin{align*} \int \frac{x^2 \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{f-g x^2} \, dx &=n \int \frac{x^2 \log (a+b x)}{f-g x^2} \, dx-n \int \frac{x^2 \log (c+d x)}{f-g x^2} \, dx-\left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right ) \int \frac{x^2}{f-g x^2} \, dx\\ &=\frac{x \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{g}+n \int \left (-\frac{\log (a+b x)}{g}+\frac{f \log (a+b x)}{g \left (f-g x^2\right )}\right ) \, dx-n \int \left (-\frac{\log (c+d x)}{g}+\frac{f \log (c+d x)}{g \left (f-g x^2\right )}\right ) \, dx-\frac{\left (f \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )\right ) \int \frac{1}{f-g x^2} \, dx}{g}\\ &=\frac{x \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{g}-\frac{\sqrt{f} \tanh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{g^{3/2}}-\frac{n \int \log (a+b x) \, dx}{g}+\frac{n \int \log (c+d x) \, dx}{g}+\frac{(f n) \int \frac{\log (a+b x)}{f-g x^2} \, dx}{g}-\frac{(f n) \int \frac{\log (c+d x)}{f-g x^2} \, dx}{g}\\ &=\frac{x \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{g}-\frac{\sqrt{f} \tanh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{g^{3/2}}-\frac{n \operatorname{Subst}(\int \log (x) \, dx,x,a+b x)}{b g}+\frac{n \operatorname{Subst}(\int \log (x) \, dx,x,c+d x)}{d g}+\frac{(f n) \int \left (\frac{\log (a+b x)}{2 \sqrt{f} \left (\sqrt{f}-\sqrt{g} x\right )}+\frac{\log (a+b x)}{2 \sqrt{f} \left (\sqrt{f}+\sqrt{g} x\right )}\right ) \, dx}{g}-\frac{(f n) \int \left (\frac{\log (c+d x)}{2 \sqrt{f} \left (\sqrt{f}-\sqrt{g} x\right )}+\frac{\log (c+d x)}{2 \sqrt{f} \left (\sqrt{f}+\sqrt{g} x\right )}\right ) \, dx}{g}\\ &=-\frac{n (a+b x) \log (a+b x)}{b g}+\frac{n (c+d x) \log (c+d x)}{d g}+\frac{x \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{g}-\frac{\sqrt{f} \tanh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{g^{3/2}}+\frac{\left (\sqrt{f} n\right ) \int \frac{\log (a+b x)}{\sqrt{f}-\sqrt{g} x} \, dx}{2 g}+\frac{\left (\sqrt{f} n\right ) \int \frac{\log (a+b x)}{\sqrt{f}+\sqrt{g} x} \, dx}{2 g}-\frac{\left (\sqrt{f} n\right ) \int \frac{\log (c+d x)}{\sqrt{f}-\sqrt{g} x} \, dx}{2 g}-\frac{\left (\sqrt{f} n\right ) \int \frac{\log (c+d x)}{\sqrt{f}+\sqrt{g} x} \, dx}{2 g}\\ &=-\frac{n (a+b x) \log (a+b x)}{b g}+\frac{n (c+d x) \log (c+d x)}{d g}+\frac{x \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{g}-\frac{\sqrt{f} \tanh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{g^{3/2}}-\frac{\sqrt{f} n \log (a+b x) \log \left (\frac{b \left (\sqrt{f}-\sqrt{g} x\right )}{b \sqrt{f}+a \sqrt{g}}\right )}{2 g^{3/2}}+\frac{\sqrt{f} n \log (c+d x) \log \left (\frac{d \left (\sqrt{f}-\sqrt{g} x\right )}{d \sqrt{f}+c \sqrt{g}}\right )}{2 g^{3/2}}+\frac{\sqrt{f} n \log (a+b x) \log \left (\frac{b \left (\sqrt{f}+\sqrt{g} x\right )}{b \sqrt{f}-a \sqrt{g}}\right )}{2 g^{3/2}}-\frac{\sqrt{f} n \log (c+d x) \log \left (\frac{d \left (\sqrt{f}+\sqrt{g} x\right )}{d \sqrt{f}-c \sqrt{g}}\right )}{2 g^{3/2}}+\frac{\left (b \sqrt{f} n\right ) \int \frac{\log \left (\frac{b \left (\sqrt{f}-\sqrt{g} x\right )}{b \sqrt{f}+a \sqrt{g}}\right )}{a+b x} \, dx}{2 g^{3/2}}-\frac{\left (b \sqrt{f} n\right ) \int \frac{\log \left (\frac{b \left (\sqrt{f}+\sqrt{g} x\right )}{b \sqrt{f}-a \sqrt{g}}\right )}{a+b x} \, dx}{2 g^{3/2}}-\frac{\left (d \sqrt{f} n\right ) \int \frac{\log \left (\frac{d \left (\sqrt{f}-\sqrt{g} x\right )}{d \sqrt{f}+c \sqrt{g}}\right )}{c+d x} \, dx}{2 g^{3/2}}+\frac{\left (d \sqrt{f} n\right ) \int \frac{\log \left (\frac{d \left (\sqrt{f}+\sqrt{g} x\right )}{d \sqrt{f}-c \sqrt{g}}\right )}{c+d x} \, dx}{2 g^{3/2}}\\ &=-\frac{n (a+b x) \log (a+b x)}{b g}+\frac{n (c+d x) \log (c+d x)}{d g}+\frac{x \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{g}-\frac{\sqrt{f} \tanh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{g^{3/2}}-\frac{\sqrt{f} n \log (a+b x) \log \left (\frac{b \left (\sqrt{f}-\sqrt{g} x\right )}{b \sqrt{f}+a \sqrt{g}}\right )}{2 g^{3/2}}+\frac{\sqrt{f} n \log (c+d x) \log \left (\frac{d \left (\sqrt{f}-\sqrt{g} x\right )}{d \sqrt{f}+c \sqrt{g}}\right )}{2 g^{3/2}}+\frac{\sqrt{f} n \log (a+b x) \log \left (\frac{b \left (\sqrt{f}+\sqrt{g} x\right )}{b \sqrt{f}-a \sqrt{g}}\right )}{2 g^{3/2}}-\frac{\sqrt{f} n \log (c+d x) \log \left (\frac{d \left (\sqrt{f}+\sqrt{g} x\right )}{d \sqrt{f}-c \sqrt{g}}\right )}{2 g^{3/2}}-\frac{\left (\sqrt{f} n\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{g} x}{b \sqrt{f}-a \sqrt{g}}\right )}{x} \, dx,x,a+b x\right )}{2 g^{3/2}}+\frac{\left (\sqrt{f} n\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{g} x}{b \sqrt{f}+a \sqrt{g}}\right )}{x} \, dx,x,a+b x\right )}{2 g^{3/2}}+\frac{\left (\sqrt{f} n\right ) \operatorname{Subst}\left (\int \frac{\log \left (1+\frac{\sqrt{g} x}{d \sqrt{f}-c \sqrt{g}}\right )}{x} \, dx,x,c+d x\right )}{2 g^{3/2}}-\frac{\left (\sqrt{f} n\right ) \operatorname{Subst}\left (\int \frac{\log \left (1-\frac{\sqrt{g} x}{d \sqrt{f}+c \sqrt{g}}\right )}{x} \, dx,x,c+d x\right )}{2 g^{3/2}}\\ &=-\frac{n (a+b x) \log (a+b x)}{b g}+\frac{n (c+d x) \log (c+d x)}{d g}+\frac{x \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{g}-\frac{\sqrt{f} \tanh ^{-1}\left (\frac{\sqrt{g} x}{\sqrt{f}}\right ) \left (n \log (a+b x)-\log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-n \log (c+d x)\right )}{g^{3/2}}-\frac{\sqrt{f} n \log (a+b x) \log \left (\frac{b \left (\sqrt{f}-\sqrt{g} x\right )}{b \sqrt{f}+a \sqrt{g}}\right )}{2 g^{3/2}}+\frac{\sqrt{f} n \log (c+d x) \log \left (\frac{d \left (\sqrt{f}-\sqrt{g} x\right )}{d \sqrt{f}+c \sqrt{g}}\right )}{2 g^{3/2}}+\frac{\sqrt{f} n \log (a+b x) \log \left (\frac{b \left (\sqrt{f}+\sqrt{g} x\right )}{b \sqrt{f}-a \sqrt{g}}\right )}{2 g^{3/2}}-\frac{\sqrt{f} n \log (c+d x) \log \left (\frac{d \left (\sqrt{f}+\sqrt{g} x\right )}{d \sqrt{f}-c \sqrt{g}}\right )}{2 g^{3/2}}+\frac{\sqrt{f} n \text{Li}_2\left (-\frac{\sqrt{g} (a+b x)}{b \sqrt{f}-a \sqrt{g}}\right )}{2 g^{3/2}}-\frac{\sqrt{f} n \text{Li}_2\left (\frac{\sqrt{g} (a+b x)}{b \sqrt{f}+a \sqrt{g}}\right )}{2 g^{3/2}}-\frac{\sqrt{f} n \text{Li}_2\left (-\frac{\sqrt{g} (c+d x)}{d \sqrt{f}-c \sqrt{g}}\right )}{2 g^{3/2}}+\frac{\sqrt{f} n \text{Li}_2\left (\frac{\sqrt{g} (c+d x)}{d \sqrt{f}+c \sqrt{g}}\right )}{2 g^{3/2}}\\ \end{align*}

Mathematica [A]  time = 0.24023, size = 467, normalized size = 0.85 \[ \frac{\sqrt{f} n \left (\text{PolyLog}\left (2,\frac{b \left (\sqrt{f}-\sqrt{g} x\right )}{a \sqrt{g}+b \sqrt{f}}\right )-\text{PolyLog}\left (2,\frac{d \left (\sqrt{f}-\sqrt{g} x\right )}{c \sqrt{g}+d \sqrt{f}}\right )+\log \left (\sqrt{f}-\sqrt{g} x\right ) \left (\log \left (\frac{\sqrt{g} (a+b x)}{a \sqrt{g}+b \sqrt{f}}\right )-\log \left (\frac{\sqrt{g} (c+d x)}{c \sqrt{g}+d \sqrt{f}}\right )\right )\right )-\sqrt{f} n \left (\text{PolyLog}\left (2,\frac{b \left (\sqrt{f}+\sqrt{g} x\right )}{b \sqrt{f}-a \sqrt{g}}\right )-\text{PolyLog}\left (2,\frac{d \left (\sqrt{f}+\sqrt{g} x\right )}{d \sqrt{f}-c \sqrt{g}}\right )+\log \left (\sqrt{f}+\sqrt{g} x\right ) \left (\log \left (-\frac{\sqrt{g} (a+b x)}{b \sqrt{f}-a \sqrt{g}}\right )-\log \left (-\frac{\sqrt{g} (c+d x)}{d \sqrt{f}-c \sqrt{g}}\right )\right )\right )-\sqrt{f} \log \left (\sqrt{f}-\sqrt{g} x\right ) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )+\sqrt{f} \log \left (\sqrt{f}+\sqrt{g} x\right ) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )-\frac{2 \sqrt{g} (a+b x) \log \left (e \left (\frac{a+b x}{c+d x}\right )^n\right )}{b}+\frac{2 \sqrt{g} n (b c-a d) \log (c+d x)}{b d}}{2 g^{3/2}} \]

Antiderivative was successfully verified.

[In]

Integrate[(x^2*Log[e*((a + b*x)/(c + d*x))^n])/(f - g*x^2),x]

[Out]

((-2*Sqrt[g]*(a + b*x)*Log[e*((a + b*x)/(c + d*x))^n])/b + (2*(b*c - a*d)*Sqrt[g]*n*Log[c + d*x])/(b*d) - Sqrt
[f]*Log[e*((a + b*x)/(c + d*x))^n]*Log[Sqrt[f] - Sqrt[g]*x] + Sqrt[f]*Log[e*((a + b*x)/(c + d*x))^n]*Log[Sqrt[
f] + Sqrt[g]*x] + Sqrt[f]*n*((Log[(Sqrt[g]*(a + b*x))/(b*Sqrt[f] + a*Sqrt[g])] - Log[(Sqrt[g]*(c + d*x))/(d*Sq
rt[f] + c*Sqrt[g])])*Log[Sqrt[f] - Sqrt[g]*x] + PolyLog[2, (b*(Sqrt[f] - Sqrt[g]*x))/(b*Sqrt[f] + a*Sqrt[g])]
- PolyLog[2, (d*(Sqrt[f] - Sqrt[g]*x))/(d*Sqrt[f] + c*Sqrt[g])]) - Sqrt[f]*n*((Log[-((Sqrt[g]*(a + b*x))/(b*Sq
rt[f] - a*Sqrt[g]))] - Log[-((Sqrt[g]*(c + d*x))/(d*Sqrt[f] - c*Sqrt[g]))])*Log[Sqrt[f] + Sqrt[g]*x] + PolyLog
[2, (b*(Sqrt[f] + Sqrt[g]*x))/(b*Sqrt[f] - a*Sqrt[g])] - PolyLog[2, (d*(Sqrt[f] + Sqrt[g]*x))/(d*Sqrt[f] - c*S
qrt[g])]))/(2*g^(3/2))

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Maple [F]  time = 0.418, size = 0, normalized size = 0. \begin{align*} \int{\frac{{x}^{2}}{-g{x}^{2}+f}\ln \left ( e \left ({\frac{bx+a}{dx+c}} \right ) ^{n} \right ) }\, dx \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(x^2*ln(e*((b*x+a)/(d*x+c))^n)/(-g*x^2+f),x)

[Out]

int(x^2*ln(e*((b*x+a)/(d*x+c))^n)/(-g*x^2+f),x)

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Maxima [F(-2)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Exception raised: ValueError} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*log(e*((b*x+a)/(d*x+c))^n)/(-g*x^2+f),x, algorithm="maxima")

[Out]

Exception raised: ValueError

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Fricas [F]  time = 0., size = 0, normalized size = 0. \begin{align*}{\rm integral}\left (-\frac{x^{2} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )}{g x^{2} - f}, x\right ) \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*log(e*((b*x+a)/(d*x+c))^n)/(-g*x^2+f),x, algorithm="fricas")

[Out]

integral(-x^2*log(e*((b*x + a)/(d*x + c))^n)/(g*x^2 - f), x)

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Sympy [F(-1)]  time = 0., size = 0, normalized size = 0. \begin{align*} \text{Timed out} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x**2*ln(e*((b*x+a)/(d*x+c))**n)/(-g*x**2+f),x)

[Out]

Timed out

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Giac [F]  time = 0., size = 0, normalized size = 0. \begin{align*} \int -\frac{x^{2} \log \left (e \left (\frac{b x + a}{d x + c}\right )^{n}\right )}{g x^{2} - f}\,{d x} \end{align*}

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(x^2*log(e*((b*x+a)/(d*x+c))^n)/(-g*x^2+f),x, algorithm="giac")

[Out]

integrate(-x^2*log(e*((b*x + a)/(d*x + c))^n)/(g*x^2 - f), x)